Biomedical Engineering Reference
In-Depth Information
is the random variable counting those tips which are born in
A
.
By definition
d
Φ
(
{
0
}×
R
)=
N
0
. This process
Φ
is characterized by the following stochastic
intensity:
d
μ
(
d
t
×
d
x
)=
prob
(
Φ
(
d
t
×
d
x
)=
1
|F
t
−
)=
Λ
t
(
d
x
)
d
t
,
for
(
t
,
x
,
s
)
∈
R
+
×
R
.
F
t
−
denotes the
σ
algebra history of the process up to time
t
−
and, given a
d
nonnegative function
h
∈
C
b
(R
)
and
α
∈
R
+
,
t
−
)
i
=
1
ε
X
i
(
t
)
(
d
x
)
.
N
(
Λ
t
(
d
x
)=
α
h
(
C
(
t
,
x
))
(9)
When a tip located in
x
branches at time
t
, the initial value of the state of the new
tip is taken as
X
N
(
t
)+
1
v
N
(
t
)+
1
(
,
)=(
x
,
v
0
)
,where
v
0
is a non random velocity and
T
N
(
t
)+
1
=
t
. Given the branching process
Φ
, the counting process
N
(
t
)=
Φ
([
0
,
t
]
×
d
R
is a Poisson-like stochastic process, counting the total number of cells born
before
t
)
∈
R
+
, with intensity
N
)
i
=
1
h
(
C
(
t
,
X
i
(
t
d
ν
(
d
t
)=
Λ
t
(R
)
d
t
=
(
t
)))
d
t
.
Evolution of the Fields.
The chemotactic field
C
diffuses and degrades; the
consumption is proportional to the extension velocities
v
i
(
t
,
x
)
,
i
=
1
,...,
N
(
t
)
.So,for
d
,
any
(
t
,
x
)
∈
R
+
×
R
N
(
t
)
i
=
1
(
v
i
∂
∂
1
N
t
C
(
t
,
x
)=
c
1
δ
A
(
x
)+
c
2
C
(
t
,
x
)
−
c
3
C
(
t
,
x
)
(
t
)
δ
X
i
)
∗
K
N
)(
x
)
.
(10)
(
t
c
3
∈
R
+
in Eq. (
10
) represent the rate of production of a source
located in a region
A
Parameters
c
1
,
c
2
,
d
, modelling, e.g., a tumor mass, the diffusivity and the
rate of consumption, respectively.
Fibronectin is known to be attached to the extracellular matrix and does not
diffuse [
2
], thus the equation for fibronectin does not contain any diffusion term.
As in [
36
], degradation of fibronectin, characterized by a coefficient
⊂
R
β
2
, depends
on the concentration of MDE, produced by the cells. Hence, the concentration of
fibronectin
f
(
x
,
t
)
produced by the endothelial cells at the tip evolves as
N
)
i
=
1
(
δ
X
i
(
t
)
∗
K
N
)(
x
)
−
β
2
m
(
t
,
x
)
f
(
t
,
x
)
,
(
t
∂
∂
)=
β
1
1
N
f
(
t
,
x
(11)
t
for
β
1
,
β
2
∈
R
+
. The MDE, once produced with rate
ν
1
, diffuses locally with
diffusion coefficient
ε
1
and is spontaneously degraded at a rate
ν
2
.
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